You might also be able to work around the problem by running the cached downloaded installer by hand, located in:

%LOCALAPPDATA%\MonoDevelop-4.0\Cache\TempDownload

(For quick reference, I double-checked the update behavior briefly in one local test Visual Studio 2015 environment, starting with Xamarin 4.6.0.297 for Visual Studio 2015, downloading the update, and then clicking the Install button. The installer ran as expected for that quick test, so I expect there are some specific environment details needed to hit the issue. The team can help investigate further on a bug report.)